Computing GCRDs of Approximate Differential Polynomials

TL;DR

This paper introduces a new algorithm for computing approximate greatest common right divisors (GCRDs) of differential polynomials with approximate coefficients, extending symbolic methods to approximate differential operators.

Contribution

It presents the first algorithm for approximate GCRDs of differential polynomials using a linearized Sylvester matrix and SVD, with implementation and robustness demonstration.

Findings

Algorithm successfully finds nearby polynomials with a non-trivial GCRD.

Implemented in Maple, demonstrating robustness.

Extends symbolic GCD methods to approximate differential operators.

Abstract

Differential (Ore) type polynomials with approximate polynomial coefficients are introduced. These provide a useful representation of approximate differential operators with a strong algebraic structure, which has been used successfully in the exact, symbolic, setting. We then present an algorithm for the approximate Greatest Common Right Divisor (GCRD) of two approximate differential polynomials, which intuitively is the differential operator whose solutions are those common to the two inputs operators. More formally, given approximate differential polynomials $f$ and $g$, we show how to find "nearby" polynomials $\widetilde f$ and $\widetilde g$ which have a non-trivial GCRD. Here "nearby" is under a suitably defined norm. The algorithm is a generalization of the SVD-based method of Corless et al. (1995) for the approximate GCD of regular polynomials. We work on an appropriately "linearized" differential Sylvester matrix, to which we apply a block SVD. The algorithm has been implemented in Maple and a demonstration of its robustness is presented.